Advertisement

Galois Extensions

  • Juliusz Brzeziński
Chapter
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Abstract

In this chapter, we arrive at the main theorems of Galois theory. Following the material covered in previous chapters, we are now equipped to define Galois extensions, i.e. the extensions which are both normal and separable. One of the most central results is Galois’ correspondence between the subgroups of the Galois group of such an extension and the intermediate subfields of it. In the exercises, we find many examples and interesting properties of Galois extensions. Several exercises are related to the inverse problem in Galois theory: to construct a Galois extension of the rational numbers with given group as its Galois group. In a series of exercises, this problem is solved for different groups of small orders and for all cyclic groups using a special case of Dirichlet’s theorem on primes in arithmetic progression, which is proved in the next chapter.

References

  1. [F]
    E. Formanek, Rational function fields. Noether’s problem and related questions. Journal of Pure and Applied Algebra, 31(1984), 28–36.CrossRefzbMATHGoogle Scholar
  2. [FK]
    E. Fried, J. Kollár, Authomorphism Groups of Algebraic Number Fields, Math. Z., 163 (1978), 121–123.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [MM]
    G. Malle, B.H. Matzat, Inverse Galois Theory, Springer, 1999.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Juliusz Brzeziński
    • 1
    • 2
  1. 1.Department of Mathematical SciencesUniversity of GothenburgGöteborgSweden
  2. 2.Chalmers University of TechnologyGöteborgSweden

Personalised recommendations